Structured time-delay models for dynamical systems with connections to Frenet-Serret frame

Area of Science:

• Dynamical systems theory
• Differential geometry
• Data-driven modeling

Background:

• Time-delay embedding and dimensionality reduction are key for representing physical system dynamics.
• The Hankel Alternative View of Koopman (HAVOK) uses dynamic mode decomposition on time-delay coordinates for linearizing nonlinear systems.
• HAVOK models exhibit approximate antisymmetric and tridiagonal matrix structures, with a forcing term for chaotic systems.

Purpose of the Study:

• Establish a novel theoretical link between HAVOK and the Frenet-Serret frame.
• Develop an improved HAVOK algorithm for more stable and accurate models from less data.
• Leverage geometric insights to enhance nonlinear system identification.

Main Methods:

• Connecting HAVOK's matrix structure to Frenet-Serret frame curvatures.
• Modifying the HAVOK algorithm to enforce antisymmetric properties.
• Applying the improved method to synthetic and real-world nonlinear system data.

Main Results:

• Demonstrated that sub- and super-diagonal entries of the HAVOK model relate to Frenet-Serret frame curvatures.
• Developed an algorithm that promotes the antisymmetric structure for improved robustness.
• Showcased enhanced modeling performance in noisy, low-data scenarios.

Conclusions:

• The geometric interpretation provides a deeper understanding of HAVOK.
• The modified algorithm offers a more reliable approach for nonlinear system identification.
• This work advances data-driven modeling by integrating differential geometry principles.